102 The Rev. J. Challis on the Analytical 



complete differential of a function of three independent varia- 

 bles, the motion of the fluid is rectilinear. This theorem, 

 when once established, becomes so essential a part of analyti- 

 cal hydrodynamics, and so materially affects much that has 

 been written in this department of science, that I make no 

 apology for adding a direct proof of it. 



Let ar, y, z be the coordinates of any point of the fluid at a 

 given time, and x + dx, y + dy, z + dz the coordinates at 

 the same time of another point distant from the former by the 

 indefinitely small line d s. Let ds make angles a, /3, y with 

 the axes of rectangular coordinates, and let the direction of 

 the velocity V at the point xyss make angles of, /3', 7' with 

 the same axes, and an angle with the line ds. Then, the 

 components of V in the directions of the axes being w, v, w, 

 we have 



, ,, , ( u dx v dy w dz\ 

 udx + vdy + wdz = Vds.(- v .— + - v . Ts + v . Ts ), 



= Vds. (cosa cos a! + cos /3 cos /3' + cosy cosy') } 



= V ds cos 0. 



If, therefore, dr be the projection of ds on the line of motion, 

 it follows that 



udx-\-vdy + wdz=zVdr. 



This equality is true whether the left-hand side be an exact 

 differential or not. Supposing it to be an exact differential, 

 we might perhaps at once assert (since V dr must be exactly 

 integrable) that V is a function of x, y, z, which varies at a 

 given instant by change of position from point to point of the 

 line of motion, but not by change of position in any direction 

 transverse to this; in other words, that V is invariable in 

 passing from point to point of the surface of displacement of 

 which u dx + v dy + 10 dz = is the differential equation. 

 But that nothing may appear to be taken for granted in a 

 question of so much importance, I proceed to prove that V 

 must be a function of this kind, in order that the three equa- 



du dv du dw dv din . . _ . 



tl0I)S Ty = d? di ~ d2 rfl = dy> ™y bG Satisfiqd ' 



When the condition of the continuity of the fluid is main- x 

 tained, the most general supposition that can be made re- 

 specting the directions of motion in an indefinitely small 

 element of the fluid, is that they are normals to a surface of 

 continued curvature, and consequently intersect at right an- 

 gles each of two focal lines situated in the planes of greatest 

 and least curvature. In the annexed diagram let Ox, Oy, 



